Advanced Stochastic Analysis Tool

🎲 Advanced Stochastic Analysis 12-in-1

📊 Markov chains · Monte Carlo · random walks · queueing · Brownian motion · more

🔄 Markov Chain (2-state)

Transition Matrix P

Initial state (0 or 1) Steps (n)

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State probabilities after n steps, stationary distribution

🔄 Markov Chain (3-state)

Transition Matrix (rows: p00 p01 p02)

Row 2: p10 p11 p12

Row 3: p20 p21 p22

Initial state (0, 1, or 2) Steps (n)

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State probabilities after n steps

🎯 Monte Carlo: Estimate π

Number of points

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Estimates π using random points in a square

🎲 Monte Carlo: Dice Roll

Number of rolls

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Distribution of sums from rolling two dice

📊 Monte Carlo: Integration

Function (x variable) Lower bound (a) Upper bound (b) Number of samples

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Monte Carlo integration of f(x) from a to b

🚶 Random Walk (1D)

Steps Probability of right (p)

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Final position, mean, variance, drift

🚶 Random Walk (2D)

Steps

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Final position (x,y), distance from origin

🌊 Brownian Motion

Time steps (n) Time horizon (T) Volatility (σ) Drift (μ)

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Final position, mean, variance, min/max

📊 M/M/1 Queue

Arrival rate (λ) Service rate (μ)

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Utilization, avg customers, avg wait time

📊 M/M/c Queue

Arrival rate (λ) Service rate (μ) Number of servers (c)

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Multi-server queue metrics

📈 Birth-Death Process

Birth rate (λ) Death rate (μ) Initial population Steps

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Population trajectory, mean, variance

⏱️ Poisson Process

Rate (λ) Time interval (T)

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Expected events, probability of k events, interarrival times

🎲 Advanced Stochastic Analysis – 12-in-1
This comprehensive tool provides essential stochastic process functions:

1. Markov Chain (2-state) – transition probabilities, n-step probabilities, stationary distribution.
2. Markov Chain (3-state) – transition probabilities, n-step probabilities.
3. Monte Carlo: Estimate π – uses random points to approximate π.
4. Monte Carlo: Dice Roll – simulates dice rolls and shows distribution.
5. Monte Carlo: Integration – numerical integration using random sampling.
6. Random Walk (1D) – simulates a 1D random walk with drift.
7. Random Walk (2D) – simulates a 2D random walk.
8. Brownian Motion – simulates geometric Brownian motion with drift.
9. M/M/1 Queue – single-server queue metrics.
10. M/M/c Queue – multi-server queue metrics.
11. Birth-Death Process – simulates population growth with birth and death rates.
12. Poisson Process – expected events, probabilities, interarrival times.

💡 Tips:
• All simulations use pseudo-random numbers – results may vary.
• All calculations are performed client-side – no data is sent anywhere.
• Perfect for students, researchers, and data analysts.

📖 Advanced Stochastic Analysis – detailed guide

🔄 Markov Chains
A Markov chain is a stochastic process with the Markov property (memoryless).
• 2-state: Transition matrix P = [[p00, p01], [p10, p11]] where each row sums to 1.
• 3-state: Transition matrix P = [[p00, p01, p02], [p10, p11, p12], [p20, p21, p22]].
• Stationary distribution π satisfies π = πP.

🎯 Monte Carlo Methods
Monte Carlo methods use random sampling to estimate numerical results.
• Estimate π: place random points in a square, count those inside a circle (π/4).
• Dice roll: simulate rolling two dice and show the distribution of sums.
• Integration: approximate ∫f(x)dx by sampling random points and averaging f(x).

🚶 Random Walks
A random walk is a stochastic process where each step is random.
• 1D: position after n steps with probability p of moving right.
• 2D: position after n steps moving in four directions (up, down, left, right).

🌊 Brownian Motion
A continuous-time stochastic process with independent increments.
• Geometric Brownian motion: dX = μX dt + σX dW.
• Simulates asset prices, particle diffusion, and more.

📊 Queueing Theory
Models waiting lines and service systems.
• M/M/1: single server, Poisson arrivals, exponential service.
• M/M/c: c servers, Poisson arrivals, exponential service.
• Key metrics: utilization ρ, average customers L, average wait time W.

📈 Birth-Death Process
A Markov process where population changes by ±1.
• Birth rate λ, death rate μ.
• The process is stable if λ < μ.

⏱️ Poisson Process
A counting process with rate λ.
• Expected events in time T: λT.
• Probability of k events: P(N(T)=k) = e^(-λT)(λT)ᵏ/k!.
• Interarrival times are exponential with mean 1/λ.

⚡ All tools work offline · perfect for students and researchers

✅ advanced stochastic analysis · 12-in-1 · dynamic · no server